3.7-3.8 Solving Systems by Substitution and by Elimination

What is the 1st step to solving a real-world application problem of systems?

• Solve the system using the Substitution or Elimination

• Write 2 linear equations to create a system

• Define the variables (x and y)

• Write answers with labels (in context of the problem)

Answer explanation

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F-LE.1

Which methods can be used to SOLVE a systems application problem?

• Graphing method

• Substitution method

• Elimination method

• None of these methods

Answer explanation

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F-LE.1

1) Define the variables for the system:

When you get insurance on your home, you have the option to add extra insurance for very valuable items like jewelry and art.

You find that your insurance company charges a flat fee plus a percentage of the value of the item.

It costs $39 to insure a $1200 and $77 to insure a $3100 item.

What are the flat fee and percentage charges?

• x = # of jewelry items

  y = # of art pieces

• m = flat rate

  b = percentage

• x = insurance premium

  y = value of item

• m = percentage rate

  b = flat fee

• x = home insurance

  y = extra insurance

Answer explanation

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F-LE.1

1) Define the variables for the system:

José and Jorge sell mango and guavas at their fruit stand.

When you arrive to sell fruit for them, you realize that they never told you what the prices were and there is no sign.

You see two orders that you can use to figure out the two unknown prices.

1 mango and three guavas cost $12, and 2 mangos and 1 guava cost $5.

How much do the mangoes and guavas cost?

• x = price of mangoes

  y = price of guavas

• x = number of mangoes

  y = number of guavas

• x = total of 1st order

  y = total of 2nd order

• x = $12

  y = $5

• x = 3 mangoes

  y = 4 guavas

Answer explanation

Tags

F-LE.1

2) WRITE 2 EQUATIONS for the SYSTEM:

When you get insurance on your home, you have the option to add extra insurance for very valuable items like jewelry and art.

You find that your insurance company charges a flat fee plus a percentage of the value of the item.

It costs $39 to insure a $1200 and $77 to insure a $3100 item.

What are the flat fee and percentage charges?

• x = 39m + 1200

  y = 77m + 3100

• 39x + 77y = 116

  1200x + 3100y = 4500

• 39x + 1200 = b

  77x + 3100 = b

• 1200 = 39m + b

  3100 = 77m + b

• 39 = 1200m + b

  77 = 3100m + b

Answer explanation

Tags

F-LE.1

2) WRITE 2 EQUATIONS for the SYSTEM:

José and Jorge sell mango and guavas at their fruit stand.

When you arrive to sell fruit for them, you realize that they never told you what the prices were and there is no sign.

You see two orders that you can use to figure out the two unknown prices.

1 mango and three guavas cost $12, and 2 mangos and 1 guava cost $5.

How much do the mangoes and guavas cost?

• x + 3y = 12

  2x + y = 5

• x + 2x = 3

  3y + y = 4

• 4x + 12 = y

  3x + 5 = y

• x = 3y + 12

  2x = y + 5

• x + 2y = 7

  3x + 4y = 17

Answer explanation

Tags

F-LE.1

3) SOLVE the SYSTEM:

When you get insurance on your home, you have the option to add extra insurance for very valuable items like jewelry and art.

You find that your insurance company charges a flat fee plus a percentage of the value of the item.

It costs $39 to insure a $1200 and $77 to insure a $3100 item.

What are the flat fee and percentage charges?

• The flat fee is $15.

  The percentage charge is 2% of the value of the item.

• The percentage charge is 15% of the value of the item.

  The flat fee is $2.

• The percentage charge is 3.25% of the value of the item.

  The flat fee is $2.48.

• The flat fee is $3.25.

  The percentage charge is 2.48% of the value of the item.

• The flat fee is $50.

  The percentage charge is 30% of the value of the item.

Answer explanation

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F-LE.1

3) SOLVE the SYSTEM:

José and Jorge sell mango and guavas at their fruit stand.

When you arrive to sell fruit for them, you realize that they never told you what the prices were and there is no sign.

You see two orders that you can use to figure out the two unknown prices.

1 mango and three guavas cost $12, and 2 mangos and 1 guava cost $5.

How much do the mangoes and guavas cost?

• Mangoes are $0.60 each.

  Guavas are $3 each.

• Mangoes are $2.50 each.

  Guavas are $4 each.

• Mangoes are $3 each.

  Guavas are $2.50 each.

• Mangoes are $3.80 each.

  Guavas are $0.60 each.

• Mangoes are $0.60 each.

  Guavas are $3.80 each.

Answer explanation

Tags

F-LE.1

Reorder the following steps for solving a system of linear equations by elimination.

• Substitute:

  Substitute your solved variable back into either equation to solve for the other variable OR repeat steps 1-4 to find the other variable.

• Solve:
  Solve for the variable that remains.

• Rearrange:

  Rearrange both equations into standard form. Line up the x and y terms in each equation.

• Add:

  Add the equations together to eliminate one variable.

• Multiply:

  Multiply one (or both) equations by a constant to get “opposites”. You want one variable with equal but opposite coefficients in each equation.

Answer explanation

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F-LE.1

Reorder the following steps for solving a system of linear equations by substitution.

• Check it:

  Check your solution by plugging into BOTH equations of the system to be sure that you get a true statement.

• Solve:

  There should only be one variable remaining.

  Solve that for that variable.

• Isolate one variable:

  Solve one equation for either variable (if necessary)

• Plug in:

  To find the other variable, plug your solution into one of the original equations and solve.

• Substitute:

  Substitute that solved equation into the other equation.

Answer explanation

Tags

F-LE.1